Timeline for what are the singularities of a normal crossings divisor?

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Jun 21, 2013 at 3:18	comment	added	Jesse Silliman		For a divisor with irreducible components having multiplicity one, we can identify it with a reducible subvariety. As a point on this subvariety, a point of intersection is singular, since the dimension of the cotangent space $\frac{m_x}{m_x^2}$ is not equal to the dimension of the subvariety. This is the same reason that an irreducible nodal curve is singular.
Jun 20, 2013 at 15:31	comment	added	Ariyan Javanpeykar		The points on $D$ are smooth as points on the ambient variety; in this example the ambient variety is some affine space, and the divisor $D$ is the union of the x-axis and y-axis. The origin is the only singular point on the divisor $D$ in this example.
Jun 20, 2013 at 14:32	comment	added	div90		Ok, so what happens is that the intersections are smooth when considered as subvarieties (in your example the point is smooth) but $D$ is not smooth itself. That's it?
Jun 20, 2013 at 13:52	history	answered	Jesse Silliman	CC BY-SA 3.0